properties of vector-valued functions

If  F=(f1,,fn)  and  G=(g1,,gn)  are vector-valued and u a real-valued function of the real variable t, one defines the vector-valued functionsPlanetmathPlanetmath F+G and uF componentwise as


and the real valued dot productMathworldPlanetmath as


If  n=3,  one my define also the vector-valued cross productMathworldPlanetmath function as


It’s not hard to verify, that if F, G and u are differentiableMathworldPlanetmathPlanetmath on an interval, so are also F+G, uF and FG, and the formulae


are valid, in 3 additionally


Likewise one can verify the following theorems.

Theorem 1.  If u is continuousMathworldPlanetmath in the point t and F in the point u(t), then


is continuous in the point t.  If u is differentiable in the point t and F in the point u(t), then the composite function H is differentiable in t and the chain ruleMathworldPlanetmath


is in .

Theorem 2.  If F and G are integrable on  [a,b],  so is also c1F+c2G, where c1,c2 are real constants, and


Theorem 3.  Suppose that F is continuous on the interval I and  cI.  Then the vector-valued function


is differentiable on I and satisfies  G=F.

Theorem 4.  Suppose that F is continuous on the interval  [a,b]  and G is an arbitrary function such that  G=F  on this interval.  Then


Theorem 2 may be generalised to

Theorem 5.  If F is integrable on  [a,b]  and  C=(c1,,cn)  is an arbitrary vector of n, then dot product CF is integrable on this interval and

Title properties of vector-valued functions
Canonical name PropertiesOfVectorvaluedFunctions
Date of creation 2013-03-22 19:02:42
Last modified on 2013-03-22 19:02:42
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Topic
Classification msc 26A42
Classification msc 26A36
Classification msc 26A24
Related topic ProductAndQuotientOfFunctionsSum